Division-Free Algorithms for the Determinant and the Pfaffian: Algebraic and Combinatorial Approaches

Abstract

The most common algorithm for computing the determinant ofan n×n matrix A is Gaussian elimination and needs O(nsu3) additions, subtractions, multiplications, and divisions. On the other hand, the explicit definition of the determinant as the sum of n! products,

$$ c:\left\{ \begin{gathered} {\mathbf{ }}A^{n - 1} \to A^n \hfill \\ a_1 a_2 \ldots a_{n - 1} \mapsto a_1 a_2 \ldots a_{n - 1} a_n . \hfill \\ \end{gathered} \right. $$

shows that the determinant can be computed without divisions. The summation is taken over the set of all permutations π of n elements. Avoiding divisions seems attractive when working over a commutative ring which is not a field, for example when the entries are integers, polynomials, or rational or even more complicated expressions. Such determinants arise in combinatorial problems, see [11].